Transactions of the American Mathematical Society

Author(s): Yuri Berest
Journal: Trans. Amer. Math. Soc. 352 (2000), 3743-3776.
MSC (1991): Primary 58F07, 35L15; Secondary 58F37, 35L25
Posted: April 18, 2000
Retrieve article in: PDF
This article is available free of charge

A lacuna of a linear hyperbolic differential operator is a domain inside its propagation cone where a proper fundamental solution vanishes identically. Huygens' principle for the classical wave equation is the simplest important example of such a phenomenon. The study of lacunas for hyperbolic equations of arbitrary order was initiated by I. G. Petrovsky (1945). Extending and clarifying his results, Atiyah, Bott and Gårding (1970-73) developed a profound and complete theory for hyperbolic operators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. In the present paper we study this problem for one remarkable class of partial differential operators with singular coefficients. These operators stem from the theory of special functions in several variables related to finite root systems (Coxeter groups). The underlying algebraic structure makes it possible to extend many results of the Atiyah-Bott-Gårding theory. We give a generalization of the classical Herglotz-Petrovsky-Leray formulas expressing the fundamental solution in terms of Abelian integrals over properly constructed cycles in complex projective space. Such a representation allows us to employ the Petrovsky topological condition for testing regular (strong) lacunas for the operators under consideration. Some illustrative examples are constructed. A relation between the theory of lacunas and the problem of classification of commutative rings of partial differential operators is discussed.

1.: L. Asgeirsson, Some hints on Huygens' principle and Hadamard's conjecture, Comm. Pure Appl. Math., 9 (1956), 307-326. MR 18:487d
2.: M.F. Atiyah, R. Bott, and L. Gårding, Lacunas for hyperbolic differential operators with constant coefficients I, Acta Math., 124 (1970), 109-189. MR 57:10252a
3.: M.F. Atiyah, R. Bott, and L. Gårding, Lacunas for hyperbolic differential operators with constant coefficients II, Acta Math., 131 (1973), 145-206. MR 57:10252b
4.: Yu.Yu. Berest, Lacunae of hyperbolic Riesz kernels and commutative rings of partial differential operators, Lett. Math. Phys. 41 (1997), 227-235. MR 98k:58103
5.: Yu.Yu. Berest, Solution of a restricted Hadamard problem on Minkowski spaces, Comm. Pure Appl. Math., 50 (1997), 1019-1052. MR 99c:35135
6.: Yu.Yu. Berest, The theory of lacunas and quantum integrable systems, in Proceedings of the Workshop The Calogero-Moser-Sutherland Model (J.-F. van Diejen and L. Vinet, Eds.), CRM Series in Mathematical Physics, Springer-Verlag (1998), to appear.
7.: Yu.Yu. Berest, and Yu.A. Molchanov, Fundamental solutions for partial differential operators with reflection group invariance, J. Math. Phys. 36(8) (1995), 4324-4339. MR 96c:35005
8.: Yu.Yu. Berest, and A.P. Veselov, Huygens' principle and Coxeter groups, Russian Math. Surveys, 48(2) (1993), 183-184. MR 94i:35110
9.: Yu.Yu. Berest, and A.P. Veselov, Hadamard's problem and Coxeter groups: new examples of Huygens' equations, Funct. Anal. Appl., 28 (1994), 3-12. MR 95h:58131
10.: Yu.Yu. Berest, and A.P. Veselov, Huygens' principle and integrability, Russian Math. Surveys, 49(6) (1994), 5-77. MR 96a:35003
11.: V.A. Borovikov, Some sufficient conditions for the absence of lacunas, Mat. Sbornik, 55 (3) (1961), 237-254. (Russian) MR 25:5284
12.: L. Boutet de Monvel, Lacunas and transmissions in Seminar on Singularities of Solutions of Linear Partial Differential Equations, Ann. of Math. Stud. 91, Princeton Univ. Press, Princeton, 1978, pp. 209-218. MR 80k:58092
13.: J. Chaillou, Les Polynômes Différentiels Hyperboliques et Leurs Perturbations Singulières, Gauthiers-Villars, Paris, 1973. MR 57:3651
14.: O.A. Chalykh, and A.P. Veselov, Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys., 126 (1990), 597-611. MR 91g:58106
15.: O.A. Chalykh, and A.P. Veselov, Integrability in the theory of the Schrödinger operator and harmonic analysis, Comm. Math. Phys., 152 (1993), 29-40. MR 94a:58160
16.: T.W. Chaundy, Hypergeometric partial differential equations, Quart. J. Math., Oxford Ser. 6 (1935), 288-303; 7 (1936), 305-315; 8 (1937), 280-302; 9 (1938), 234-240; 10 (1939), 219-240; 11 (1940), 101-110. MR 1:119g
17.: R. Courant, and D. Hilbert, Methods of Mathematical Physics II, Interscience Publ., N. Y., 1962. MR 25:4216
18.: G.F.D. Duff, Singularities, supports and lacunas, in Advances in Microlocal Analysis, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 168, Reidel, Dordrecht-Boston, 1986, pp. 73-133. MR 87j:35003
19.: J.J. Duistermaat, and L. Hörmander, Fourier integral operators II, Acta Math., 128 (1972), 183-269. MR 52:9300
20.: C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. AMS, 311 (1989), 167-183. MR 90k:33027
21.: C.F. Dunkl, Hankel transforms associated to finite reflection groups, Contemporary Math., 138 (1992), 123-138. MR 94g:33011
22.: F.G. Friedlander, The Wave Equation on a Curved Space-Time, Cambridge Univ. Press, Cambridge, 1975.MR 57:889
23.: A.M. Gabrielov, and A.P. Palamodov, Huygens' principle and its generalizations in I.G.Petrovsky ''Selected Works``, Nauka, Moscow, 1986, pp. 449-456.MR 88f:01059; English transl. in: I. G. Petrovsky, ``Selected Works'', Part I, Classics of Soviet Mathematics 5, Gordon and Breach Publ., Amsterdam, 1996, pp. 485-495.
24.: S.A. Gal'pern, and V.E. Kondrashov, The Cauchy problem for differential operators decomposing into wave factors, Trans. Moscow Math. Soc., 16 (1967), 117-145.MR 37:585
25.: L. Gårding, The solution of Cauchy's problem for two totally hyperbolic linear differential equations by means of Riesz integrals, Ann. Math., 48 (1947), 785-826. MR 9:240a
26.: L. Gårding, Linear hyperbolic partial differential equations with constant coefficients, Acta Math., 85 (1950), 1-62. MR 12:831g
27.: L. Gårding, Sharp fronts of paired oscillatory integrals, Publ. RIMS, Kyoto Univer., 12 (1977), 53-68 (Corrections: ibid., 13 (1977), 821). MR 57:10253a; MR 57:10253b
28.: L. Gårding, Singularities in Linear Wave Propagation, Lecture Notes in Math, 1241, Springer-Verlag, Berlin, 1987. MR 88k:35002
29.: I.M. Gel'fand, and G.E. Shilov, Generalized Functions I, Academic Press, New York, 1964. MR 29:3869
30.: S.G. Gindikin, Analysis in homogeneous domains, Russ. Math. Surveys, 19(4) (1964), 1-89. MR 30:2167
31.: S.G. Gindikin, Tube Domains and the Cauchy Problem, Transl. Math. Monographs, 111, Amer. Math. Soc., Providence, RI, 1992. MR 94d:35028
32.: P. Günther, Huygens' Principle and Hyperbolic Equations, Academic Press, Boston, 1988. MR 89j:35077
33.: J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale Univ. Press, New Haven, 1923 (reprinted by Dover, N. Y., 1956).MR 14:474f
34.: J. Hadamard, The problem of diffusion of waves, Annals of Math., 43(3) (1942), 510-522. MR 4:45a
35.: G.J. Heckman, A remark on the Dunkl differential-difference operators, Progr. in Math., 101 Birkhäuser, Boston, 1991, pp. 181-191. MR 94c:20075
36.: G.J. Heckman, and H.Schlichtkrull, Harmonic Analysis and Special Functions on Symmetric Spaces, Academic Press, San Diego, 1994. MR 96j:22019
37.: S. Helgason, Wave equations on homogeneous spaces, Lecture Notes in Math., 1077 (1984), 252-287. MR 86c:58141
38.: L. Hörmander, Fourier integral operators I, Acta Math., 127 (1971), 79-183. MR 52:9299
39.: L. Hörmander, The Analysis of Linear Partial Differential Operators, I-IV, Springer-Verlag, Berlin, 1983-85. MR 85g:35002a, MR 85g:35002b; MR 87d:35002a; MR 87d:35002b
40.: I.M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations, Russian Math. Surveys, 32(6) (1977), 198-220. MR 58:24353
41.: J.E. Lagnese, A solution of Hadamard's problem for a rectricted class of operators, Proc. Amer. Math. Soc., 19 (1968), 981-988. MR 37:6581
42.: A. Lax, On Cauchy's problem for partial differential equations with multiple characteristics, Comm. Pure Appl. Math., 9 (1956), 135-169. MR 18:397b
43.: H. Lewy, The wave equation as limit of hyperbolic equations of higher order, Comm. Pure Appl. Math., 17 (1965), 5-16. MR 30:5059
44.: M. Mathisson, Le probléme de M. Hadamard relatif a la diffusion des ondes, Acta Math., 71 (1939), 249-282. MR 1:120e
45.: R.G. McLenaghan, Huygens' principle, Ann. Inst. Henri Poincaré Sect. A 37(3) (1982), 211-236. MR 84e:35091
46.: W. Nuij, A note on hyperbolic polynomials, Math. Scand., 23 (1968), 69-72. MR 40:3368
47.: M.A. Olshanetsky, and A.M. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Reps., 94 (1983), 313-404. MR 84k:81007
48.: E. Opdam Root systems and hypergeometric functions III, IV, Compositio Math., 67 (1988), 21-49, 191-209. MR 90k:17012; MR 90c:58079
49.: E. Opdam, Dunkl operators, Bessel functions, and the discriminant of a finite Coxeter group, Compositio Math., 85 (1993), 333-373. MR 95j:33044
50.: I.G. Petrowsky, On the diffusion of waves and the lacunas for hyperbolic equations, Mat. Sbornik, 17(59) (1945), 289-370. (English; Russian summary) MR 8:79a
51.: M. Riesz, L'intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math., 81 (1949), 1-223. MR 10:713c
52.: K.L. Stellmacher, Ein Beispiel einer Huyghensschen Differentialgleichung, Nachr. Akad. Wiss. Göttingen, Math.-Phys. K1, IIa, 10 (1953), 133-138.MR 15:710e
53.: K.L. Stellmacher, Ein Klasse Huyghenscher Differentialgleichung und ihre Integration, Math. Ann., 130(3) (1955), 219-233. MR 17:494e
54.: S.L. Svensson, Necessary and sufficient conditions for the hyperbolicity of polynomials with hyperbolic principal part, Arkiv Math., 8 (1969), 145-162 MR 42:6421
55.: B.R. Vainberg, S.G. Gindikin, A strengthened Huygens' principle for a class of differential operators with constant coefficients, Trans. Moscow Math. Soc., 16 (1967), 163-196. MR 37:3176
56.: V.A. Vassiliev, Sharpness and local Petrovsky condition for strictly hyperbolic operators with constant coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 50(2) (1986), 243-284; English transl., Math. USSR Izv. 28 (1987), 233-273. MR 88a:58193
57.: V.A. Vassiliev, Ramified Integrals, Singularities and Lacunas, Mathematics and its Appl., 315, Kluwer Acad. Publ., 1995. MR 96h:32052
58.: A.P. Veselov, K.L. Styrkas, and O.A. Chalykh, Algebraic integrability for the Schrödinger operator and reflection groups, Theor. Math. Phys. 94 (1993), 182-197. MR 94j:35151
59.: A.P. Veselov, M.V. Feigin, and O.A. Chalykh, New integrable generalizations of quantum Calogero-Moser problem, Usp. Mat. Nauk, 51(3) (1996), 185-186; English transl. in Russian Math. Surveys 57 (1996), no. 3. MR 97e:58122

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58F07, 35L15, 58F37, 35L25

Yuri Berest
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Department of Mathematics, Cornell University, Ithaca, New York 14853-4201
Email: berest@math.cornell.edu

DOI: 10.1090/S0002-9947-00-02543-5
PII: S 0002-9947(00)02543-5
Keywords: Hyperbolic linear differential operators, fundamental solution, lacuna, Huygens' principle, Coxeter groups, Dunkl operators
Received by editor(s): March 6, 1997
Received by editor(s) in revised form: November 3, 1997
Posted: April 18, 2000
Copyright of article: Copyright 2000, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS